How to remove these strange lines?
It's easy. Your UV map islands overlap with each other, and with the image border. Consider using some gaps for your UV map. Also you can use some bleeding in the paint Options tab to extend filling even more.
Also ensure if the 'Subdivide UVs' in the subdiv modifier is checked.
And also you can disable subdiv modifier when you paint the texture. The paint mode considers subdivided UVs, hence it keeps some edges uncovered.